Justice Department regulations call for Mueller's report to be a confidential account of the individuals charged, as well as those who were not charged. Barr, who was confirmed this month, will then summarise the work for Congress.
Schiff also took aim at Barr, saying that if the new attorney-general withholds any part of the report, his legacy will be "tarnished." Ultimately, the Democratic lawmaker added, US President Donald Trump should welcome the report's release, given his repeated claims that there was no collusion between his campaign and the Russian Government.
"We are going to share this information with the public, and if the President is serious about all of his claims of exoneration, then he should welcome the publication of this report," Schiff said.
Opinions on whether Democrats would be able to successfully subpoena Mueller were split.
In an appearance on NBC News, Neal Katyal, who was acting solicitor general in the Obama Administration, said it was "certainly possible" that Mueller would testify before Congress.
If Mueller ends up finding information that points to potential wrongdoing by the President, Barr not only has the discretion to turn the report over to Congress, "indeed, he has to," said Katyal, who drafted the special counsel regulations.
"The overall intent of the regulations - it's said time and time again - is public confidence in the administration of justice. And any sort of suppressed report about presidential wrongdoing will flunk that test," he said.
On CBS News, Senator Roy Blunt, R, voiced uncertainty about whether Democrats could enforce a subpoena against Mueller.
"I don't know that you can," he said.
Blunt also declined to say whether the Senate Intelligence Committee, of which he is a member, may pursue its own effort to have Mueller testify. "I think we'll have to wait and see what's in the report," he said.
If Trump asks Barr to let him read the report, that is "perfectly fine," said Solomon Wisenberg, who was deputy independent counsel in the probe of President Bill Clinton.
But the problem will be if Trump orders Barr to take a certain action on the report, Wisenberg said.
"I don't think Barr will stand for that," Wisenberg said.